It is fairly common for accommodation providers of all kinds to refuse to do a TM30, including some major hotels. But that explanation won't be accepted at Immigration, IF they ask for the TM30. It's possible that you'll need a Plan B.
Odd that your prospective employer could not provide guidance on this. Are you certain that the employer has provided all the relevant documentation? Close coordination with them will be needed.
Whatever visa you have, you need to politely make it very visible and/or say what you've got. You can't just hand over your passport and play the "find my visa" lottery.
Be aware that you cannot obtain a tourist visa in-country, you will have to leave Thailand. You cannot switch from a Non-B to a tourist visa at immigration.
As you say, it is impossible to know for certain. Having a visa is not infallible, and it's not a guarantee - that applies to any country.
If you were questioned heavily, as I and many others have been in recent months, the decision as to how seriously you treat this red flag is entirely yours, but it IS a red flag.
You need to start pushing the school to do its job and guide you through this. If they are letting you flail around guessing that you can use a 15 year-old police check then they could seriously mess you about later on. I know they probably brushed you off and said "go do on website" but they have a responsibility to guide you through this.